Question: What is the greatest common divisor of $5!$ and $\frac{8!}{3!}?$ Express your answer as an integer.  (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.)
Answer: First, we factor $5!$ into primes: \begin{align*} 5! &= 5\cdot4\cdot3\cdot2\cdot1\\ &= 2^{3}\cdot3\cdot5. \end{align*} Then, we factor $\frac{8!}{3!}.$ \begin{align*} \frac{8!}{3!} &= 8\cdot7\cdot6\cdot5\cdot4\\ &= 2^{6}\cdot 3\cdot 5\cdot 7. \end{align*} We can find the greatest common factor by taking the lowest exponent of each common prime factor. We get $2^{3}\cdot3\cdot5 = \boxed{120}.$